Constraints on muon-specific dark forces

The recent measurement of the Lamb shift in muonic hydrogen allows for the most precise extraction of the charge radius of the proton which is currently in conflict with other determinations based on $e-p$ scattering and hydrogen spectroscopy. This discrepancy could be the result of some new muon-specific force with O(1–100) MeV force carrier—in this paper we concentrate on vector mediators. Such an explanation faces challenges from the constraints imposed by the $g-2$ of the muon and electron as well as precision spectroscopy of muonic atoms. In this work we complement the family of constraints by calculating the contribution of hypothetical forces to the muonium hyperfine structure. We also compute the two-loop contribution to the electron parity-violating amplitude due to a muon loop, which is sensitive to the muon axial-vector coupling. Overall, we find that the combination of low-energy constraints favors the mass of the mediator to be below 10 MeV and that a certain degree of tuning is required between vector and axial-vector couplings of new vector particles to muons in order to satisfy constraints from muon $g-2$. However, we also observe that in the absence of a consistent standard model embedding high-energy weak-charged processes accompanied by the emission of new vector particles are strongly enhanced by $(E/m_V{)}^2$, with $E$ a characteristic energy scale and $m_V$ the mass of the mediator. In particular, leptonic $W$ decays impose the strongest constraints on such models completely disfavoring the remainder of the parameter space.

Figure 1

Left: the effective proton-muon interaction resulting from unexpectedly large QCD effects or new physics that is responsible for the $r_p$ discrepancy. Right: the two-loop contribution to the muon $g-2$ that results from the interaction on the left after integrating out the proton.

Figure 2

One-loop diagram (plus all possible crossings) contributing at $1/m_V$ order to the muonium HFS.

Figure 3

Two-loop diagram with the closed muon loop contributing to the atomic PNC amplitude. This diagram does not decouple in the large $m_{\mu }$ limit.

Figure 4

Typical representatives of muon pair production by electron-proton collision due to a new force. The parity violation will come about due to the presence of the $g_A$ coupling in the interference with the pure QED diagrams.

Figure 5

Diagram that leads to the decay $W\to \mu \nu V$.

Figure 6

Parameter space of the model, when $\kappa$ is chosen as a function of $m_V$ to saturate the $a_e$ constraint. Solid curves are limits on $g_V$, while dashed ones are limits on $g_A$. The upper green shaded band shows the range of values of $g_V$ that alleviate the $r_p$ discrepancy. The lower green shaded band shows values of $g_A$ required so that $(g-2{)}_{\mu }$ theory and experiment agree to $2\sigma$, given values of $g_V$ in the upper band. We show constraints on $g_V$ from muonium HFS, muonic Si and Mg, and $W\to \mu \nu V$ decays (solid curves) and on $g_A$ from PNC in ${}^{133}\mathrm{Cs}$ (dashed curve).

Figure 7

$R(m_V)$ as defined in Eq. (a5) over the relevant range of $m_V$.